This invention relates to modulation coding and partial response systems.
In modulation coding, symbols are encoded as signals drawn from a constellation in such a way that only certain sequences of signals are possible.
In recent years, a number of kinds of trellis-type modulation codes have been developed and applied (e.g., in modems) to realize coding gains of 3 to 6 dB over high-signal-to-noise-ratio, band-limited channels such as voice grade telephone channels.
Early trellis codes were due to Ungerboeck (Cjsaka et al., U.S. Pat. No. 3,877,768; Ungerboeck, "Channel Coding with Multilevel/Phase Signals," IEEE Transactions on Information Theory, Vol. IT-28, pp. 55-67, January 1982). Ungerboeck's codes for sending n bits per symbol are based on 4-subset or 8-subset partitions of one-dimensional (PAM) or two-dimensional (QAM) 2.sup.n+1 -point signal constellations, combined with a rate-1/2 or rate-2/3 linear binary convolutional code that determines a sequence of subsets. A further set of "uncoded" bits then determines which signal points within the specified subsets are actually sent. The partition and the code are designed to guarantee a certain minimum squared distance d.sub.min.sup.2 between permissible sequences of signal points. Even after giving effect to the power cost of an expanded signal constellation (a factor of four (6 dB) in one dimension, or a factor of two (3 dB) in two dimensions), the increase in minimum squared distance yields a coding gain that ranges from about a factor of two (3 dB) for simple codes up to a factor of four (6 dB) for the most complicated codes, for values of n that may be as large as desired.
Gallager (U.S. Pat. No. 4,755,998, continuation of U.S. application Ser. No. 577,044, filed Feb. 6, 1984, discussed in Forney et al., "Efficient Modulation for Band-Limited Channels," IEEE J. Select. Areas Commun., Vol. SAC-2, pp. 632-647, 1984) devised a multidimensional trellis code based on a 16-subset partition of a four-dimensional signal constellation, combined with a rate-3/4 convolutional code. The four-dimensional subset is determined by selecting a pair of two-dimensional subsets, and the points of the four-dimensional signal constellation are made up of pairs of points from a two-dimensional signal constellation. With only an 8-state code, a d.sub.min.sup.2 of four times the uncoded minimum sequence distance can be obtained, while the loss due to expanding the signal constellation can be reduced to about a factor of 2.sup.1/2 (1.5 dB), yielding a net coding gain of the order of 4.5 dB. A similar code was designed by Calderbank and Sloane ("Four-dimensional Modulation With An Eight-State Trellis Code", AT&T Tech. J., Vol. 64, pp. 1005- 1018, 1985; U.S. Pat. No. 4,581,601).
Wei (U.S. patent application Ser. No. 727,398, filed Apr. 25, 1985, now allowed as U.S. Pat. No. 4,713,817) devised a number of multidimensional codes based on partitions of constellations in four, eight, and sixteen dimensions, combined with rate-(n-1)/n convolutional codes. His multidimensional constellations again consist of sequences of points from two-dimensional constituent constellations. The codes are designed to minimize two-dimensional constellation expansion, to obtain performance (coding gain) versus code complexity over a broad range, and for other advantages such as transparency to phase rotations. Calderbank and Sloane, "New Trellis Codes", IEEE Trans. Inf. Theory, to appear March, 1987; "An Eight-dimensional Trellis Code," Proc. IEEE, Vol. 74, pp. 757-759, 1986 have also devised a variety of multidimensional trellis codes, generally with similar performance versus complexity, more constellation expansion, but in some cases fewer states.
All of the above codes are designed for channels in which the principal impairment (apart from phase rotations) is noise, and in particular for channels with no intersymbol interference. The implicit assumption is that any intersymbol interference introduced by the actual channel will be reduced to a negligible level by transmit and receive filters; or, more specifically, by an adaptive linear equalizer in the receiver. Such a system is known to work well if the actual channel does not have severe attenuation within the transmission bandwidth, but in the case of severe attenuation ("nulls" or "near nulls") the noise power may be strongly amplified in the equalizer ("noise enhancement").
A well-known technique for avoiding such "noise enhancement" is to design the signaling system for controlled intersymbol interference rather than no intersymbol interference. The best-known schemes of this type are called "partial response" signaling schemes (Forney, "Maximum Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference," IEEE Trans. Inform. Theory, Vol. IT-18, pp. 363-378, 1972).
In a typical (one-dimensional) partial response scheme, the desired output y.sub.k at the receiver is designed to be the difference of two successive inputs x.sub.k, i.e., y.sub.k =x.sub.k -x.sub.k-1, rather than y.sub.k =x.sub.k. In sampled-data notation using the delay operator D, this means that the desired output sequence y(D) equals x(D)(1-D) rather than x(D); this is thus called a "1-D" partial response system. Because the spectrum of a discrete-time channel with impulse response 1-D has a null at zero frequency (DC), the combination of the transmit and receive filters with the actual channel likewise must have a DC null to achieve this desired response. On a channel which has a null or a near null at DC, a receive equalizer designed for a 1-D desired response will introduce less noise enhancement than one designed to produce a perfect (no intersymbol interference) response.
Partial response signaling is also used to achieve other objectives, such as reducing sensitivity to channel impairments near the band edge, easing filtering requirements, allowing for pilot tones at the band edge, or reducing adjacent-channel interference in frequency-division multiplexed systems.
Other types of partial response systems include a 1+D system which has a null at the Nyquist band edge, and a 1-D.sup.2 system which has nulls at both DC and the Nyquist band edge. A quadrature (two-dimensional) partial response system (QPRS) can be modeled as having a two-dimensional complex input; the (complex) response 1+D results in a QPRS system which has nulls at both the upper and lower band edges in a carrier-modulated (QAM) bandpass system. All of these partial response systems are closely related to one another, and schemes for one are easily adapted to another, so one can design a system for the 1-D response, say, and easily extend it to the others.
Calderbank, Lee, and Mazo ("Baseband Trellis Codes with A Spectral Null at Zero"; submitted to IEEE Trans. Inf. Theory) have proposed a scheme to construct trellis-coded sequences that have spectral nulls, particularly at DC, a problem that is related to the design of partial response systems, even though its objectives are in general somewhat different. Calderbank et al. have adapted known multidimensional trellis codes with multidimensional signal constellations to produce signal sequences with spectral nulls by the following technique. The multidimensional signal constellation has twice as many signal points as are necessary for the non-partial-response case, and is divided into two equal size disjoint subsets, one of multidimensional signal points whose sum of coordinates is less than or equal to zero, the other whose sum is greater than or equal to zero. A "running digital sum" (RDS) of coordinates, initially set to zero, is adjusted for each selected multidimensional signal point by the sum of its coordinates. If the current RDS is nonnegative, then the current signal point is chosen from the signal subset whose coordinate sums are less than or equal to zero; if the RDS is negative, then the current signal point is chosen from the other subset. In this way the RDS is kept bounded in a narrow range near zero, which is known to force the signal sequence to have a spectral null at DC. At the same time, however, the signal points are otherwise chosen from the subsets in the same way as they would have been in a non-partial-response system: the expanded multidimensional constellation is divided into a certain number of subsets with favorable distance properties, and a rate-(n-1)/n convolutional code determines a sequence of the subsets such that the minimum squared distance between sequences is guaranteed to be at least d.sub.min.sup.2. The coding gain is reduced by the constellation doubling (by a factor of 2.sup.1/2, or 1.5 dB, in four dimensions, or by a factor of 2.sup.1/4, or 0.75 dB in eight), but otherwise similar performance is achieved as in the non-partial response case with similar code complexity.